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\(\mathcal D\)-bundles and integrable hierarchies. (English) Zbl 1253.14020
This article is on the geometry of \(\mathcal{D}\)-bundle, i.e., locally projective \(\mathcal{D}\)-modules, on algebraic curves, and applications of them to the studies of integrable hierarchies, i.e., the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies, and especially to the rational, trigonometric, and elliptic solutions of KP.
Since the local structure of the \(\mathcal{D}\)-bundles is determined by the full Sato Grassmannian, the authors show that the KP hierarchies have a geometric description as flows on moduli space of the \(\mathcal{D}\)-bundles. These solutions are captured by the \(\mathcal{D}\)-bundles on cubic curves \(E\), i.e., irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1.
Fourier-Mukai transform describing the \(\mathcal{D}\)-modules on the cubic curves \(E\) in terms of (complexes of) sheaves on a twisted cotangent bundle \(E^\natural\) over \(E\) shows the correspondence between the \(\mathcal{D}\)-bundles and the CM spectral sheaves.
It implies that the poles of the KP-solutions are identified with the positions of the CM particles. They also discuss their relations to the Hitchin integrable system.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14H20 Singularities of curves, local rings
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14H52 Elliptic curves
17B65 Infinite-dimensional Lie (super)algebras
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